-1 If you mean to say Adele's lyrics only use that form because it's required by the rhyme (and by implication that in any other circumstances the phrasing would be story to tell) then I think you are wrong. If you didn't actually mean to convey that impression, I don't see this answer is addressing the question anyway.
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FumbleFingersMar 15 '12 at 16:42

I didn't mean to say anything other than what was written on the bottom of my post: "I ascribe to the rhyming theory." Language should be analyzed in context; song lyrics often bend grammar rules and sentence structure to create rhymes. "He lives, He lives, salvation to impart" may come from a beloved hymn, but the sentence structure has much room for improvement - except that the last syllable is supposed to rhyme with the soon-to-come "heart," so we let it go. As to whether or not my post answers the question, I could've posted yet another Ngram (yawn!), but I opted for a humorous reply.
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J.R.Mar 15 '12 at 22:19

Shouldn't that be subscribe? I would have edited but for my reverence for someone with twice my rep.
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TimLymingtonDec 20 '13 at 22:09

There is a slight difference. I'd suggest "no story to be told" implies that others would not consider her story worth telling, whereas "to tell" implies she thinks she has no story worth telling.
Hmm, I need to express that better.

I think "no story to be told" more frequently implies "not suitable for present company" than "not worth telling".
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jwpat7Mar 15 '12 at 16:46

The best way I can think of to "express that better" is to paraphrase OP's first version as implying "no story you would think is interesting enough to be worth hearing", and the second as implying "no story I think is 'objectively' interesting enough to be worth telling". An exceptionally fine distinction, if it could in fact be made at all. But I don't think many if any speakers would ever choose one over the other on that basis.
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FumbleFingersMar 15 '12 at 16:49

Good points FF, but sometimes it's exactly those nuances that make the opening of a book/song/poem work. I enjoy thinking about them anyway
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WudangMar 15 '12 at 20:31

The common English usage would be 'I have no story to tell'. If anything, 'no story to be told' is more of an old English phrasing; but I agree in the lyric it is used for rhyming purposes. Technically, both are correct.

It's purely a matter of stylistic preference in OP's context - there's no difference at all in meaning. Statistically, to tell is a little more common, but only because it's the simpler verb form. To be told would never be thought of as even slightly uncommon or unusual. In a context like...

Dad says you were a sailor, Uncle Jack! I bet you have a story to tell!

...nearly everyone would use to tell, but I can't think of any corresponding context where nearly everyone would use to be told.

I see a difference in that I have no story to tell could be a broader statement of "there is no information on any particular topic that I deem worthy of relating", whereas I have no story to be told is much more specifically about how "there is nothing about me that is worth the time it would take to tell you about it."

To take the opposite of each statement, if I have a story to tell, it could be a story about anybody. If I have a story to be told, it can only be a story about me.